Package gvcm.cat. March 16, 2015

Transcription

1 Type Package Package gvcm.cat March 16, 2015 Title Regularized Categorical Effects/Categorical Effect Modifiers/Continuous/Smooth Effects in GLMs Version 1.9 Date Author Margret Oelker Maintainer Margret Oelker Description Generalized structured regression models with regularized categorical effects, categorical effect modifiers, continuous effects and smooth effects. Depends R (>= ), Matrix, MASS, splines, mgcv License GPL (>= 2) LazyLoad yes NeedsCompilation no Repository CRAN Date/Publication :26:36 R topics documented: cat_control gvcm.cat gvcm.cat-internal gvcm.cat.flex index plot.gvcm.cat predict.gvcm.cat simulation Index 17 1

2 2 cat_control cat_control Auxiliary Function for gvcm.cat Description Usage Auxiliary function for gvcm.cat. Modifies the algorithm s internal parameters. cat_control(center = FALSE, standardize = FALSE, accuracy = 2, digits = 4, g = 0.5, epsilon = 10^(-5), maxi = 250, c = 10^(-5), gama = 20, steps = 25, nu = 1, tuning.criterion = "GCV", K = 5, cv.refit = FALSE, lambda.upper=50, lambda.lower=0, lambda.accuracy=.01, scaled.lik=false, adapted.weights=false, adapted.weights.adj = FALSE, adapted.weights.ridge = FALSE, assured.intercept=true, level.control = FALSE, case.control = FALSE, pairwise = TRUE, grouped.cat.diffs = FALSE, bootstrap = 0, start.ml = FALSE, L0.log = TRUE, subjspec.gr = FALSE, high = NULL,...) Arguments center standardize accuracy digits logical; if TRUE, all metric covariates are centered by their empirical mean logical; if TRUE, the design matrix is standardized by its (weighted) empirical variances integer; number of digits being compared when setting coefficents equal/to zero integer; number of digits for estimates g step length parameter for the PIRLS-algorithm; out of )0,1( epsilon maxi c gama steps small, positive, real constant; the PIRLS-algorithm is terminated when the (scaled, absolute) difference of the coefficients of the current iteration and the coefficients of the previous iteration is smaller than epsilon integer; maximal number of iterations in the fitting algorithm small, positive, real constant; needed for the approximation of the absolute value function in the PIRLS-algorithm positive number; tuning parameter for the approximation of the L0 norm integer; tuning parameter for path-plotting; minimal number of estimates employed for path-plotting nu optional weighting parameter tuning.criterion loss criterion for cross-validation; one out of "GCV" (generalized cross validation criterion), "deviance" (K-fold cross-validation with the predictive deviance as criterion) K cv.refit integer; number of folds for cross-validation logical; if TRUE, cross-validation is based on a refit of the selected coefficients

3 cat_control 3 lambda.upper lambda.lower integer; upper bound for cross-validation of lambda integer; lower bound for cross-validation of lambda lambda.accuracy numeric; how accurate shall lambda be cross-validated?; minimal absolute difference between two candidates for lambda scaled.lik if TRUE, the likelihood in the objective function is scaled by 1/n adapted.weights logical; if TRUE, penalty terms are weighted adaptively, that is by inverse MLestimates; set to FALSE, if ML-estimates do not exist/are to close to zero; only for specials v, p, grouped, SCAD, elastic adapted.weights.adj logical; if TRUE, adapted weights of several categorical covariates are scaled such that they are comparable adapted.weights.ridge logical; if TRUE, adapted weights are based on aa estimate that is slightly penalized by a Ridge penalty assured.intercept logical; shall a constant intercept remain in the model in any case? level.control case.control pairwise logical; if TRUE, the penalty terms are adjusted for different number of penalty terms per covariate logical; if TRUE, the penalty terms are adjusted for the number of observations on each level of a categorical covariate experimental option; disabled if TRUE grouped.cat.diffs experimental option; disabled if FALSE bootstrap experimental option; disabled if 0 start.ml L0.log subjspec.gr high logical; if TRUE, the initial value is the ML-estimate experimental option; disabled if TRUE experimental option; disabled if FALSE experimental option; disabled if NULL... further arguments passed to or from other methods Value Returns a list containing the (checked) input arguments. See Also Function gvcm.cat

4 4 gvcm.cat gvcm.cat Regularized Categorical Effects/Categorical Effect Modifiers/Continuous/Smooth effects in GLMs Description Usage The function fits generalized linear models with regularized categorical effects, categorical effect modifiers, continuous effects and smooth effects. The model is specified by giving a symbolic description of the linear predictor and a description of the error distribution. Estimation employs different regularization and model selection strategies. These strategies are either a penalty or a forward selection strategy employing AIC/BIC. For non-differentiable penalties, a local quadratic approximation is employed, see Oelker and Tutz (2013). gvcm.cat(formula, data, family = gaussian, method = c("lqa", "AIC", "BIC"), tuning = list(lambda=true, specific=false, phi=0.5, grouped.fused=0.5, elastic=0.5, vs=0.5, spl=0.5), weights, offset, start, control, model = FALSE, x = FALSE, y = FALSE, plot=false,...) pest(x, y, indices, family = gaussian, tuning = list(lambda=true, specific=false, phi=0.5, grouped.fused=0.5, elastic=0.5, vs=0.5, spl=0.5), weights, offset, start = NULL, control = cat_control(), plot=false,...) abc(x, y, indices, family = gaussian, tuning = c("aic", "BIC"), weights, offset, start, control = cat_control(), plot=false,...) Arguments formula data family method tuning an object of class formula: a symbolic description of the model to be fitted. See details a data frame, containing the variables in the model a family object describing the error distribution and link function to be used in the model; this can be a character string naming a family function, a family function or the result of a call to a family function, see family for details; currently only gaussian, binomial, poisson, Gamma are working fitting method; one out of "lqa", "AIC" or "BIC"; method "lqa" induces penalized estimation; it employs a PIRLS-algorithm (see Fan and Li, 2001; Oelker and Tutz, 2013). Methods "AIC" and "BIC" employ a forward selection strategy a list; tuning parameters for penalized estimation; lambda is the scalar, overall penalty parameter; if lambda is a vector of values, these values are crossvalidated; if lambda = TRUE, lambda is cross-validated on log scale between lambda.lower and lambda.upper; see cat_control. If lambda is a vector with the same length as elements in the formula and if specific equals a vector of proper length, the entries of specific are interpreted as specific tuning

5 gvcm.cat 5 weights offset start control model parameters for each entry of the formula. phi, grouped.fused, elastic, vs and spl are parameters that weigh the terms of some penalties; must be out of intervall )0,1(; the default 0.5 corresponds to equal weights an optional weight vector (for the observations) an optional offset initial values for the PIRLS algorithm for method lqa a list of parameters for controlling the fitting process; if emtpy, set to cat_control(); see cat_control for functions gvcm.cat: a logical value indicating whether the employed model frame shall be returned or not x, y for function gvcm.cat: logical values indicating whether the response vector and the model matrix used in the fitting process shall be returned or not; for functions pest and abc: y must be a response vector, x a proper coded design matrix plot indices logical; if TRUE, estimates needed to plot coefficient paths are computed for pest and abc only: the to be used index argument; see function index... further arguments passed to or from other methods Details A typical formula has the form response ~ 1 + terms; where response is the response vector and terms is a series of terms which specifies a linear predictor. There are some special terms for regularized terms: v(x, u, n="l1", bj=true) : varying coefficients enter the formula as v(x,u) where u denotes the categorical effect modifier and x the modfied covariate. A varying intercept is denoted by v(1,u). Varying coefficients with categorical effect modifiers are penalized as described in Oelker et. al The argument bj and the element phi in argument tuning allow for the described weights. p(u, n="l1") : ordinal/nominal covariates u given as p(u) are penalized as described in Gertheiss and Tutz (2010). For numeric covariates, p(u) indicates a pure Lasso penalty. grouped(u,...): penalizes a group of covariates with the grouped Lasso penalty of Yuan and Lin (2006); so far, working for categorical covariates only sp(x, knots=20, n="l2"): implents a continuous x covariate non-parametrically as f(x); f(x) is represented by centered evaluations of basis functions (cubic B-splines with number of knots = knots); for n="l2", the curvature of f(x) is penalized by a Ridge penalty; see Eilers and Marx (1996) SCAD(u): penalizes a covariate u with the SCAD penalty by Fan and Li (2001); for categorical covariates u, differences of coefficients are penalized by a SCAD penalty, see Gertheiss and Tutz (2010) elastic(u): penalizes a covariate u with the elastic net penalty by Zou and Hastie (2005); for categorical covariates u, differences of coefficients are penalized by the elastic net penalty, see Gertheiss and Tutz (2010)

6 6 gvcm.cat Value If the formula contains no (varying) intercept, gvcm.cat assumes a constant intercept. There is no way to avoid an intercept. For specials p and v, there is the special argument n: if n="l1", the absolute values in the penalty are replaced by squares of the same terms; if n="l2", the absolute values in the penalty are replaced by quadratic, Ridge-type terms; if n="l0", the absolute values in the penalty are replaced by an indicator for non-zero entries of the same terms. For methods "AIC" and "BIC", the coefficients are not penalized but selected by a forward selection strategy whenever it makes sense; for special v(x,u), the selection strategy is described in Oelker et. al. 2012; the approach for the other specials corresponds to this idea. For binomial families the response can also be a success/failure rate or a two-column matrix with the columns giving the numbers of successes and failures. Function pest computes penalized estimates, that is, it implements method "lqa" (PIRLS-algorithm). Function abc implements the forward selection strategy employing AIC/BIC. Categorical effect modifiers and penalized categorical covariates are dummy coded as required by the penalty. If x in v(x,u) is binary, it is effect coded (first category refers to -1). Other covariates are coded like given by getoption. There is a summary function: summary.gvcm.cat gvcm.cat returns an object of class gvcm.cat which inherits from class glm which inherits from class lm. An object of class gvcm.cat contains: coefficients named vector of coefficients coefficients.reduced reduced vector of coefficients; selected coefficients/differences of coefficients are set to zero coefficients.refitted refitted vector of coefficients; i.e. maximum likelihood estimate of that model containing selected covariates only; same length as coefficients.reduced coefficients.oml maximum likelihood estimate of the full model residuals fitted.values rank deviance residuals fitted mean values degrees of freedom model; for method="lqa" estimated by the trace of the generalized head matrix; for methods "AIC", "BIC" estimated like default in glm.fit family the family object used linear.predictors linear fit on link scale deviance aic null.deviance scaled deviance a version of Akaike s Information Criterion; minus twice the maximized loglikelihood plus twice the rank. For binomial and Poison families the dispersion is fixed at one. For a gaussian family the dispersion is estimated from the residual deviance, and the number of parameters is the rank plus one. the deviance for the null model, comparable with deviance; the null model includes a non-varying intercept only

7 gvcm.cat 7 Note iter weights df.residual df.null converged boundary offset control contrasts na.action plot tuning number of iterations working weights of the final iteration the residual degrees of freedom/degrees of freedom error; computed like rank the residual degrees of freedom for the null model logical; fulfills the PIRLS-algorithm the given convergence conditions? logical; is the fitted value on the boundary of the attainable values? the offset vector used the value of the control argument used the contrasts used information returned by model.frame on the special handling of NAs; currently always na.omit in principle, a list containing two matrixes needed for different types of plots: if input option plot=true, the first matrix contains estimates needed to plot coefficient paths; if lambda was cross-validated, the second matrix contains the cross-validation scores a list, employed tuning parameters; if lambda was cross-validated, the optimal value is returned indices used index argument; see function index number.selectable.parameters number of coefficients that could be selected number.removed.parameters number of actual removed coefficients x.reduction a matrix; transforms model frame x into its reduced version; e.g. needed for refitting beta.reduction a matrix; transforms the coefficients into its reduced version call formula terms data the matched call the formula supplied the terms object used the data argument x, y if requested, the model matrix/the response vector model if requested, the model frame xlevels a record of the levels of the factors used in fitting bootstrap.errors experimental method same as input argument method In addition, non-empty fits will have components qr, R and effects relating to the final weighted linear fit. Pleas note that the functions gvcm.cat, pest and the fitting procedure for penalized estimation gvcmcatfit are organized like the functions glm/glm.fit whenever possible. This was done to avoid mistakes and to provide a well-known structure.

8 8 gvcm.cat Author(s) Margret-Ruth Oelker References Eilers, P. H. C. and B. D. Marx (1996). Flexible smoothing with b-splines and penalties. Statist. Sci. 11 (2), Fan, J. and R. Li (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456), Gertheiss, J. and G. Tutz (2010). Sparse modeling of categorial explanatory variables. The Annals of Statistics 4(4), Oelker, M.-R., J. Gertheiss and G. Tutz (2012). Regularization and model melection with categorial predictors and effect modifiers in generalized linear models. Department of Statistics at the University of Munich: Technical Report 122. Oelker, M.-R., J. Gertheiss and G. Tutz (2013). A general family of penalties for combining differing types of penalties in generalized structured models. Department of Statistics at the University of Munich: Technical Report 139. Yuan, M. and Y. Lin (2006). Model selection and estimation in regression with grouped variables. R. Stat. Soc. Ser. B Stat. Methodol. 68 (1), Zou, H. and T. Hastie (2005). Regularization and variable selection via the Elastic Net. R. Stat. Soc. Ser. B Stat. Methodol. 67 (2), See Also Functions index, cat_control, plot.gvcm.cat, predict.gvcm.cat, simulation Examples ## example for function simulation() covariates <- list(x1=list("unif", c(0,2)), x2=list("unif", c(0,2)), x3=list("unif", c(0,2)), u=list("multinom",c(0.3,0.4,0.3), "nominal") ) true.f <- y ~ 1 + v(x1,u) + x2 true.coefs <- c(0.2, 0.3,.7,.7, -.5) data <- simulation(400, covariates, NULL, true.f, true.coefs, binomial(), seed=456) ## example for function gvcm.cat() f <- y ~ v(1,u) + v(x1,u) + v(x2,u) m1 <- gvcm.cat(f, data, binomial(), plot=true, control=cat_control(lambda.upper=19)) summary(m1) ## example for function predict.gvcm.cat newdata <- simulation(200, covariates, NULL, true.f, true.coefs, binomial(), seed=789) prediction <- predict.gvcm.cat(m1, newdata) ## example for function plot.gvcm.cat plot(m1) plot(m1, type="score") plot(m1, type="coefs")

9 gvcm.cat-internal 9 gvcm.cat-internal Internal Function of gvcm.cat() Description For internal use only. See Also Function gvcm.cat gvcm.cat.flex Regularized Effects with Flexible Smoothing Parameters Description The function fits the same models with the same approximation as in gvcm.cat but the choice of the tuning parameter lambda for the penalty differs: instead of weighting the penalty terms and choosing on global tuning parmeter based on (generalized) cross-validation methods that again rely on the converged model, gvcm.cat.flex estimates several penalty parameteres lambda_i by linking the local quadratic approximation of gvcm.cat with the fantastic methods implemented in the package mgcv. This is why the arguments of gvcm.cat and gvcm.cat.flex differ. gvcm.cat.flex is not as well-developed as gvcm.cat. Usage gvcm.cat.flex(whichcoefs, intercept = TRUE, data, family = gaussian(), method = "REML", tuning = NULL, indexnrcoefs, indexpennorm, indexpena, indexpenweight, control = list(c=1e-05, epsilon=1e-07, gama=35, maxi=1500, nu=.5)) Arguments whichcoefs vector with covariates (as characters) intercept logical data a data frame, with named and coded covariates family a family object describing the error distribution and link function to be used in the model; see family for details; everyl family that is compatible with gam is working method see gam tuning for function gam: argument sp indexnrcoefs vector with number of coefficients per covariate indexpennorm vector with norm of the employed penalty (as.character) indexpena list with the penalty matrices A_j for each covariate j indexpenweight list, possible weights for the penalty terms (each entry is a vector) control a list of parameters for controlling the fitting process; must be NULL or contain all named elements

10 10 gvcm.cat.flex Details Value The local quadratic approximation are linked to the methods of mgcv by alternating the update of the penalty and the update of the PIRLS algorithm/estimating the tuning parameters lambda_i via mgcv. Therefore, gvcm.cat.flex can be slow (but will be faster than gvcm.cat for the most part). See Also A gamobject. Function gvcm.cat. Examples ## Not run: # compare gvcm.cat.flex and gvcm.cat for Lasso-type penalties: n <- 100 ncov <- 7 set.seed(123) X <- matrix(rnorm(n*ncov, sd=5), ncol=ncov) coefs <- rpois(ncov + 1, 2) y <- cbind(1, X) data <- as.data.frame(cbind(y, X)) names(data) <- c("y", paste("x", 1:ncov, sep="")) m1 <- gvcm.cat.flex( whichcoefs = paste("x", 1:ncov, sep=""), data=data, indexnrcoefs=rep(1, ncov), indexpennorm=rep("l1", ncov), indexpena=list(1,1,1,1,1,1,1), indexpenweight=list(1,1,1,1,1,1,1) ) m2 <- gvcm.cat(y ~ 1 + p(x1) + p(x2) + p(x3) + p(x4) + p(x5) + p(x6) + p(x7), data=data, tuning=list(lambda=m1$sp, specific=true), start=rep(1, 8)) rbind(m1$coefficients, m2$coefficients) # Lasso-type fusion penalty with gvcm.cat.flex n <- 100 ncat <- 8 set.seed(567) X <- t(rmultinom(n, 1, rep(1/ncat, ncat)))[, -1] coefs <- c(rpois(1, 2), sort(rpois(ncat-1, 1))) y <- cbind(1, X) data <- as.data.frame(y) data$x1 <- X names(data) <- c("y", "x1")

11 index 11 A <- a(1:(ncat-1), ncat-2) m3 <- gvcm.cat.flex( whichcoefs = c("x1"), data = data, indexnrcoefs = c(ncat-1), indexpennorm = c("l1"), indexpena = list(a), indexpenweight = list(rep(1, ncol(a))), tuning = 100 # fixed and large - in order to demonstrate the fusion of the coefficients ) m3$coefficients ## End(Not run) index Functions to Build Design Matrices and Indices for Function gvcm.cat() Description Usage design() builds design matrices for function gvcm.cat; index() computes indices with information about the terms of the formula. design(formula, data) index(dsgn, data = data, formula = formula) Arguments formula data dsgn an object of class "formula"; see gvcm.cat a data frame; see gvcm.cat value of function design() Details Function index returns a matrix with one indicator vector per row. The columns refer to the elements of the formula (same order). The indicator/indices are: index1 : gives the number of coefficients belonging to each term in the formula. An entry is 1 if the according term is metric, it equals the number of the coded variable s categories, if the variable is a factor. If a continuous variable is modified by a factor u the entry equals the number of u s categories index2 : indicates varying coefficients. An entry is 0 if the according coefficient is not varying, it is -1 if the according coefficient is nominal, 1 if it is ordinal

12 12 index Value index2b: conforms to indicator b j in Oelker et. al index3 : indicates penalized covariates p(u). An entry is 0 if the according covariate is not penalized, it is -1 if the according covariate is nominal, 1 if it is ordinal or metric index4 : indicates penalized covariates grouped(u). An entry is 0 if the according covariate is not penalized, it is -1 if the according covariate is nominal, 1 if it is ordinal or metric index5 : experimental index6 : indicates penalized covariates sp index7 : indicates penalized covariates SCAD. An entry is 0 if the according covariate is not penalized, it is -1 if the according covariate is nominal, 1 if it is ordinal or metric index8 : indicates penalized covariates elastic. An entry is 0 if the according covariate is not penalized, it is -1 if the according covariate is nominal, 1 if it is ordinal or metric index9 : experimental X Terms m int formula a matrix the model matrix the according terms.object the model frame either 0, indicating that the intercept is varying, or 1 indicating that the intercept is constant sorted version of the given formula, index vectors will refer to this formula value of function index References Oelker, M.-R., J. Gertheiss and G. Tutz (2012). Regularization and model melection with categorial predictors and effect modifiers in generalized linear models. Department of Statistics at the University of Munich: Technical Report 122. See Also Functions pest, abc Examples ## example for function simulation() covariates <- list(x1=list("unif", c(0,2)), x2=list("unif", c(0,2)), x3=list("unif", c(0,2)), u=list("multinom",c(0.3,0.4,0.3), "nominal") ) true.f <- y ~ 1 + v(x1,u) + x2 true.coefs <- c(0.2, 0.3,.7,.7, -.5) data <- simulation(400, covariates, NULL, true.f, true.coefs, binomial(), seed=456) ## example for function index() f <- y ~ v(1,u) + v(x1,u) + v(x2,u)

13 plot.gvcm.cat 13 dsgn <- design(f, data) index(dsgn, data) plot.gvcm.cat Plot Method for gvcm.cat Objects Description Function to visualize a gvcm.cat object. Usage ## S3 method for class gvcm.cat plot(x, accuracy = 2, type = "path", individual = FALSE, xlim, ylim, main = NULL, indent = 0, color = TRUE, xscale = "lambda", label = TRUE, intercept = TRUE,...) Arguments x accuracy type individual xlim ylim main indent color a gvcm.cat object; for type="path", a gvcm.cat object with value plot unequal NA is required integer; number of digits being compared when setting coefficents equal/to zero for plotting one out of "path", "score", "coefs"; defines the type of the plot logical; for type="path" and type="coefs" only; for type="path", it indicates whether the paths of all coefficients shall be plotted into one common figure (default) or in an individual figure per covariate; paths of single covariates can be selected by giving a vector containing the covariates (as characters and as given in the formula, e.g.: individual.paths=c("v(1,u)", "v(x1,u1)")) for type="coefs", the default is one plot per covariate. individual allows to select single covariates. the x limits (x1, x2) of the plot the y limits (y1, y2) of the plot title of the plot numeric; if larger zero, coefficient names printed on top of each other are adjusted logical; if FALSE, lines are gray and dotted/dashed xscale for type="path" only; if xscale="lambda", the x-axis is scaled as 1 λ/λ max ; if xscale="beta", the scale of the x-axis is the scaled L1 norm of the penalized coefficients. label intercept omits addtional information printed in the plot, if FALSE for type="coefs" and type="path" only; if FALSE, for type="path", the path of the intercept is not plotted; if FALSE, for type="coefs", intercept is not added to smooth functions... further arguments passed to or from other methods

14 14 predict.gvcm.cat Details Default option type="path" delivers a graphic with the coefficient paths between 0 (= maximal penalization) and 1 (= no penalization). Maximal penalization is defined by the minimal penalty parameter lambda that sets all penalized coefficients to zero (to constant relating to the intercept and assured.intercept = TRUE). Minimal penalization means no penalization at all, i.e. lambda = 0. Of course the minimal penalty parameter causing maximal penalization depends on how selection and clustering of coefficients is defined (see function gvcm.cat and cat_control). Coefficients belonging to one covariate are plotted in the same color, coefficients that are not modified are plotted as dashed lines. Paths are drawn by connecting steps estimates related to different values of lambda, see cat_control. Option type="score" plots the cross-validation score (depending on criterion in cat_control) as a function of penalty parameter lambda and marks the chosen penalty parameter as a dotted line. Opton type="coefs" plots the penalized coefficients whenever possible. So far, there is no plot for methods "AIC" and "BIC". Value A plot. See Also Function gvcm.cat Examples ## see example for function gvcm.cat predict.gvcm.cat Predict Method for gvcm.cat Fits Description Obtains predictions from a fitted gvcm.cat object. Usage ## S3 method for class gvcm.cat predict(object, newdata, type = "link",...) Arguments object newdata type a fitted object of class gvcm.cat a data frame in which to look for variables with which to predict the type of prediction required. The default is on the scale of the linear predictors; the alternative "response" is on the scale of the response variable. Thus for a binomial model the default predictions are of log-odds (probabilities on logit scale) and type = "response" gives the predicted probabilities... further arguments passed to or from other methods

15 simulation 15 Details Value Observations containing NAs are always omitted. fit fit.refitted fit.oml na.action predictions predictions assuming refitted coefficients predictions assuming maximum likelihood estimates information returned by model.frame on the special handling of NAs; currently always na.omit See Also Function gvcm.cat Examples ## see example for function gvcm.cat simulation Simulates data with categorial covariates Description Usage Simulates data with categorial covariates/categorial effect modifiers simulation(n, covariates, correlation = NULL, formula, coefficients, family, sd = 1, seed = rpois(1, 2348) * rnorm(1)) Arguments n covariates correlation formula coefficients family sd seed number of observations; must be large enough, so that all categories of all factor variables exist and therefore vector coefficients fits description of the covariates and effect modifiers included in the model; format: list( name of variable 1 = list("distribution", c(parameters), "level of measurement" optional matrix, specifies the correlation of Gaussian covariates formula like in gvcm.cat (all variables contained in formula must be defined in covariates) true parameter vector a family object; currently only gaussian, binomial, poisson, Gamma if family = gaussian, standard deviation of response; if family = Gamma the rate parameter like in rgamma specifies the to be used seed

16 16 simulation Details Value Remarks on covariates: all parameterizations like default in Distributions. possible distributions of covariates (required as characters), their parameters (required as vectors) and constraints (in parentheses): beta : shape1 (>0), shape2 (>0) exp : rate (>0) gamma : shape (>0) lnorm : mean, sd (>0) multinom: vector of the categories probabilities (all elements must be >0, sum over all elements must be 1) norm : mean, sd (>0) pois : lambda (>0) unif : min, max level of measurement is only needed for distribution = "multinom", must be "nominal" or "ordinal". If any, the covariates correlation is specified by argument correlation. Correlations are defined for Gaussian covariates only. Matrix correlation refers to these covariates according to the order they are listed in covariates. So that the dimensions of correlation must fit to the number of normal distributed variables in covariates. A data frame containing all specified covariates (even if they are not included in formula) and the response (named y) See Also Function gvcm.cat Examples ## example function simulation covariates <- list(x1=list("unif", c(0,2)), x2=list("unif", c(0,2)), x3=list("unif", c(0,2)), u=list("multinom",c(0.3,0.4,0.3), "nominal") ) true.f <- y ~ 1 + v(x1,u) + x2 true.coefs <- c(0.2, 0.3,.7,.7, -.5) data <- simulation(400, covariates, NULL, true.f, true.coefs, binomial(), seed=456)

17 Index Topic gvcm.cat cat_control, 2 index, 11 plot.gvcm.cat, 13 predict.gvcm.cat, 14 simulation, 15 a (gvcm.cat-internal), 9 abc, 12 abc (gvcm.cat), 4 abc.a.coefs (gvcm.cat-internal), 9 abcfit (gvcm.cat-internal), 9 bootstrap (gvcm.cat-internal), 9 cat_control, 2, 4, 5, 8, 14 check.simulation (gvcm.cat-internal), 9 contr.effect (gvcm.cat-internal), 9 cv.lambda (gvcm.cat-internal), 9 cv.vectors (gvcm.cat-internal), 9 design (index), 11 Distributions, 16 elastic (gvcm.cat-internal), 9 family, 4, 6, 9, 15 formula, 4 7, 15, 16 gam, 9 gamobject, 10 getoption, 6 glm, 6, 7 glm.fit, 6, 7 grouped (gvcm.cat-internal), 9 gvcm.cat, 2, 3, 4, 9 11, gvcm.cat-internal, 9 gvcm.cat.flex, 9, 9, 10 gvcmcatfit (gvcm.cat-internal), 9 gvcmcatfitridge (gvcm.cat-internal), 9 index, 5, 7, 8, 11 lm, 6 mgcv, 9, 10 model.frame, 7, 15 NA, 7, 15 na.omit, 7, 15 p (gvcm.cat-internal), 9 path.matrix (gvcm.cat-internal), 9 pest, 12 pest (gvcm.cat), 4 plot.gvcm.cat, 8, 13 predict.gvcm.cat, 8, 14 print.gvcm.cat (gvcm.cat), 4 pspline (gvcm.cat-internal), 9 reduce (gvcm.cat-internal), 9 rgamma, 15 SCAD (gvcm.cat-internal), 9 simulation, 8, 15 sp (gvcm.cat-internal), 9 summary.gvcm.cat (gvcm.cat), 4 terms, 7 terms.object, 12 v (gvcm.cat-internal), 9 vspline (gvcm.cat-internal), 9 weight.function (gvcm.cat-internal), 9 17